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Projector Approach to the Butuzov–Nefedov Algorithm for Asymptotic Solution of a Class of Singularly Perturbed Problems in a Critical Case

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Abstract

Under some conditions, an asymptotic solution containing boundary functions of two types was constructed by V.F. Butuzov and N.N. Nefedov for initial value problems for differential equations involving the second power of a small parameter multiplying the derivative with a right-hand side consisting of a singular matrix \(A(t)\) times the unknown function (as a linear part of the equation) plus the same small parameter multiplying a nonlinear function. In the present paper, an algorithm for constructing asymptotics using the orthogonal projectors onto \(\text{ker}A(t)\) and \(\text{ker}A(t)'\) (the prime denotes transposition) is given. This approach can be useful for understanding the algorithm underlying the construction of asymptotics. It allows us to present formulas for finding asymptotic terms of any order in an explicit form.

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ACKNOWLEDGMENTS

We are deeply grateful to V.F. Butuzov for helpful discussion of this paper.

Funding

The work of G.A. Kurina was supported by the Russian Science Foundation, grant 17-11-01220. The work of N.T. Hoai was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant 101.02-2017.314.

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Authors and Affiliations

  1. Voronezh State University, 394018, Voronezh, Russia

    G. A. Kurina

  2. Voronezh Institute of Law and Economics, 394042, Voronezh, Russia

    G. A. Kurina

  3. Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, 119333, Moscow, Russia

    G. A. Kurina

  4. University of Science, Vietnam National University, Hanoi, Vietnam

    N. T. Hoai

Authors
  1. G. A. Kurina
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  2. N. T. Hoai
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Corresponding authors

Correspondence to G. A. Kurina or N. T. Hoai.

Additional information

Dedicated to Professor V.F. Butuzov on the occasion of his 80th birthday

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Kurina, G.A., Hoai, N.T. Projector Approach to the Butuzov–Nefedov Algorithm for Asymptotic Solution of a Class of Singularly Perturbed Problems in a Critical Case. Comput. Math. and Math. Phys. 60, 2007–2018 (2020). https://doi.org/10.1134/S0965542520120076

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